Connections and L_{\infty} liftings of semiregularity maps
aa r X i v : . [ m a t h . AG ] F e b CONNECTIONS AND L ∞ LIFTINGS OF SEMIREGULARITY MAPS
EMMA LEPRI AND MARCO MANETTI
Abstract.
Let E ∗ be a finite complex of locally free sheaves on a complex manifold X . Weprove that to every connection of type (1 ,
0) on E ∗ it is canonically associated an L ∞ morphism g : A , ∗ X ( H om ∗O X ( E ∗ , E ∗ )) A ∗ , ∗ X A ≥ , ∗ X [2]that lifts the 1-component of Buchweitz-Flenner semiregularity map. An application to defor-mations of coherent sheaves on projective manifolds is given. Introduction
In the remarkable paper [7], Buchweitz and Flenner generalise the classical Severi-Kodaira-Bloch’s semiregularity map to any coherent sheaf F of locally finite projective dimension on anarbitrary complex space X . Their description involves exterior powers of the cotangent complexand when X is a smooth complex manifold it reduces to a map σ : Ext X ( F , F ) → Y q ≥ H q +2 ( X, Ω qX )defined in terms of the Atiyah class At( F ) ∈ Ext X ( F , F ⊗ Ω X ) of F , see also [21]. More precisely,the Yoneda pairingExt iX ( F , F ⊗ Ω iX ) × Ext jX ( F , F ⊗ Ω jX ) → Ext i + jX ( F , F ⊗ Ω i + jX ) , ( a, b ) a ◦ b, allows to define the exponential of the opposite of the Atiyah classexp( − At( F )) ∈ Y q ≥ Ext qX ( F , F ⊗ Ω qX ) . Since X is assumed smooth, every coherent sheaf F has locally finite projective dimension andthen there are defined the trace mapsTr : Ext iX ( F , F ⊗ Ω jX ) → H i ( X, Ω jX ) , i, j ≥ . It is worth recalling that, as proved by Atiyah for vector bundles and by Illusie in the generalcase [3, 18], when X is a projective manifold, then with respect to the Hodge decomposition incohomology, the trace of the above exponential gives the Chern character of F , cf. [7, p.137]:ch( F ) = Tr(exp( − At( F ))) . The Buchweitz-Flenner semiregularity map is defined by the formula: σ = X q ≥ σ q : Ext X ( F , F ) → Y q ≥ H q +2 ( X, Ω qX ) , σ ( c ) = Tr(exp( − At( F )) ◦ c ) . The name semiregularity is motivated by the fact that when F = O Z for a locally completeintersection Z of codimension q , then the map σ q − is the same as the classical semiregularitymap defined by Bloch [6].The semiregularity map is important both in the variational Hodge conjecture and in defor-mation theory: we refer to [6] for a discussion about the first subject, while for the second werecall that Ext X ( F , F ) is the obstruction space for the functor of deformations of F . Moreover, Date : February 11, 2021.2020
Mathematics Subject Classification.
Key words and phrases. connections, Atiyah class, L-infinity maps, semiregularity. a classical result by Mukai and Artamkin [2, 10, 17], asserts that the 0th component of thesemiregularity map σ : Ext X ( F , F ) → H ( X, O X ) , σ ( c ) = Tr( c ) , annihilates obstructions; in other words the kernel of the trace map Tr : Ext X ( F , F ) → H ( X, O X )is an obstruction space for F .One of the main results of [7] is that if the Hodge to de Rham spectral sequence of X degenerates at E , then every curvilinear (and also semitrivial, see [25]) obstruction is annihilatedby semiregularity map. However Buchweitz and Flenner left unanswered the question of whetherthe semiregularity map annihilates every obstruction.As suggested in [7], and then clarified in [9, 16, 28], the correct way to interpretate thesemiregularity map is as the obstruction map of a morphism of deformation theories, with targetin a product of (formal) intermediate Jacobians: this implies in particular that from the pointof view of deformation theory it is more appropriate, for every q , to consider the composition τ q : Ext X ( F , F ) σ q −→ H q +2 ( X, Ω qX ) = H ( X, Ω qX [ q ]) i q −→ H ( X, Ω ≤ qX [2 q ]) , where Ω ≤ qX = ( ⊕ qi =0 Ω iX [ − i ] , ∂ ) is the truncated holomorphic de Rham complex and i q is inducedby the inclusion of complexes Ω qX [ q ] ⊂ Ω ≤ qX [2 q ]. Clearly, when the Hodge to de Rham spectralsequence of X degenerates at E the second map is injective and therefore τ q and σ q have thesame kernel.Finally, the paper [28] by Pridham contains a proof that the maps τ q annihilate every obstruc-tion, for every coherent sheaf on a smooth manifold, while the analogous result for the classicalBloch semiregularity map was previously proved, under some mild additional assumption, in [16].Pridham works in the framework of homotopy homogeneous functors in the category of simpli-cial commutative algebras in order to construct a morphism of deformation theories inducingthe semiregularity map.The aim of this paper is to construct, by elementary methods, an explicit morphism of defor-mation theories lifting τ in the framework of differential graded Lie algebras. We expect thatthis explicit approach should works also for the lifting of τ q , q >
1, although this appears, atthis moment, extremely complicated from the computational point of view.Roughly speaking, by a nowadays well established theory, the category of deformation prob-lems over a field of characteristic 0 is equivalent to the homotopy category of DG-Lie algebrasover the same field: two DG-Lie algebras are homotopy equivalent if they are quasi-isomorphicor, equivalently, if they have isomorphic L ∞ minimal models, and every morphism in the homo-topy category can be represented by an L ∞ morphism. The passage from homotopy classes ofDG-Lie algebras to the standard, and more geometric, description of deformation problems isgiven by taking solutions of the Maurer-Cartan equation modulus gauge action, see e.g. [25, 26]and references therein.It is well known that the deformation theory of a coherent sheaf F is controlled by R Hom( F , F )(see e.g. [14, 27]), considered as an element in the homotopy category of differential graded Liealgebras: if F admits a finite locally free resolution (e.g. if X is a smooth projective manifold)(1.1) 0 → E − n → · · · → E → F → R Hom( F , F ) is given by the Dolbeault complex A , ∗ X ( H om ∗O X ( E ∗ , E ∗ )).Recall that the quasi-isomorphic complexes E ∗ and F have the same Atiyah class and that thehypercohomology of Ω ≤ X [2] is computed by the truncated de Rham complex A ∗ , ∗ X /A ≥ , ∗ X [2]. Asin the case of locally free sheaves (see e.g. [1, 3]), the Atiyah class of E ∗ can be computed byusing connections of type (1 , Theorem 1.1 (=Corollary 4.1) . Let F be a coherent sheaf on a complex manifold X admittinga locally free resolution E ∗ as in (1.1) . Then the choice of a connection of type (1 , on E i forevery i gives an explicit lifting of τ : Ext ∗ X ( F , F ) → H ∗ ( X, Ω ≤ X [2]) ONNECTIONS AND L ∞ LIFTINGS OF SEMIREGULARITY MAPS 3 to an L ∞ morphism g : A , ∗ X ( H om ∗O X ( E ∗ , E ∗ )) A ∗ , ∗ X A ≥ , ∗ X [2] . In the above theorem, by the term liftings we mean that the linear component g : A , ∗ X ( H om ∗O X ( E ∗ , E ∗ )) → A ∗ , ∗ X A ≥ , ∗ X [2]is a morphism of complexes inducing τ in cohomology. A brief review of L ∞ morphisms betweendifferential graded Lie algebras will be given in Section 3. Theorem 1.2 (=Corollary 4.2) . Let F be a coherent sheaf on a complex manifold X admittinga locally free resolution. Then every obstruction to the deformation of F belongs to the kernel ofthe map τ : Ext X ( F , F ) → H ( X, Ω ≤ X [2]) . If the Hodge to de Rham spectral sequence of X degenerates at E , then every obstruction to thedeformation of F belongs to the kernel of the map σ : Ext X ( F , F ) → H ( X, Ω X ) , σ ( a ) = − Tr(At( F ) ◦ a ) . It is worth pointing out that our approach is almost entirely algebraic and that the abovecorollary holds also over every smooth separated scheme of finite type over a field of characteristic0 [22]: a proof of this purely algebraic analogous result is outlined in Section 5.1.1.
Notation. If V = ⊕ V i is either a graded vector space or a graded sheaf, we denote by v the degree of a homogeneous element v ∈ V . For every integer p the symbol [ p ] denotes the shiftfunctor, defined by V [ p ] i = V p + i .For a complex manifold X , its de Rham complex is denoted by ( A ∗ , ∗ X , d = ∂ + ∂ ) and theirsubcomplexes of the Hodge filtrations by A ≥ p, ∗ X . The holomorphic de Rham complex of X isdenoted by (Ω ∗ X = ⊕ i ≥ Ω iX [ − i ] , ∂ ).2. Connections and Atiyah classes
The theory of connections of type (1 ,
0) on holomorphic vector bundles [1, 3, 19] extendswithout difficulty to every complex of locally free sheaves. For simplicity of exposition we considerhere only the case of finite complexes, which is completely sufficient for our applications.Let X be a complex manifold and let E ∗ : 0 → E p δ −→ E p +1 δ −→ · · · δ −→ E q → , p, q ∈ Z , δ = 0 , be a fixed finite complex of locally free sheaves of O X -modules. We denote by H om ∗O X ( E ∗ , E ∗ )the graded sheaf of O X -linear endomorphisms of E ∗ : H om ∗O X ( E ∗ , E ∗ ) = M i H om i O X ( E ∗ , E ∗ ) , H om i O X ( E ∗ , E ∗ ) = Y j H om O X ( E j , E i + j ) . Then H om ∗O X ( E ∗ , E ∗ ) is a sheaf of locally free DG-Lie algebras over O X , with bracket equal tothe graded commutator and differential given by f [ δ, f ] = δf − ( − f f δ . For every a, b, r denote by A a,bX ( E r ) ≃ A a,bX ⊗ O X E r the sheaf of differential forms of type ( a, b )with coefficients in E r , and by ∂ : A a,bX ( E r ) → A a,b +1 X ( E r ) , ∂ ( φ · e ) = ∂ ( φ ) · e the Dolbeault differential.We consider A ∗ , ∗ X ( E ∗ ) = M a,b,r A a,bX ( E r ) EMMA LEPRI AND MARCO MANETTI as a graded sheaf on X , where the elements of A a,bX ( E r ) have degree a + b + r . When we write φ · e ∈ A ∗ , ∗ X ( E ∗ ) we always mean that φ is a differential form and e is a holomorphic section of E ∗ , or a germ of one.In accordance with the Koszul sign rule, the differential δ can be extended to a differential δ : A a,bX ( E r ) → A a,bX ( E r +1 ) , δ ( φ · e ) = ( − φ φ · δ ( e ) . We have that ∂ = δ = 0 and [ ∂, δ ] = ∂δ + δ∂ = 0, so that ∂ + δ is a differential in A ∗ , ∗ X ( E ∗ ).Therefore the space of C -linear morphisms of sheavesHom ∗ C ( A ∗ , ∗ X ( E ∗ ) , A ∗ , ∗ X ( E ∗ ))carries a natural structure of differential graded associative algebra: the product is given bycomposition and the differential is the graded commutator with ∂ + δ .Denoting by A a,bX ( − ) the global sections of A a,bX ( − ) we have two differential graded subalgebras A , ∗ X ( H om ∗O X ( E ∗ , E ∗ )) ⊂ A ∗ , ∗ X ( H om ∗O X ( E ∗ , E ∗ )) ⊂ Hom ∗ C ( A ∗ , ∗ X ( E ∗ ) , A ∗ , ∗ X ( E ∗ )) , where for ω, η ∈ A ∗ , ∗ X , f ∈ H om ∗O X ( E ∗ , E ∗ ) , e ∈ E ∗ one has that ( ω · f )( η · e ) = ( − fη ( ω ∧ η ) · f ( e ) , so that the elements of A a,bX ( H om n O X ( E ∗ , E ∗ )) have degree a + b + n . For every a ∈ A ∗ , ∗ X ( H om ∗O X ( E ∗ , E ∗ ))we have(2.1) ∂a = [ ∂, a ]where the bracket on the right is intended in the DG-Lie algebra Hom ∗ C ( A ∗ , ∗ X ( E ∗ ) , A ∗ , ∗ X ( E ∗ )). Infact, for ω, η ∈ A ∗ , ∗ X , f ∈ H om ∗O X ( E ∗ , E ∗ ) and e ∈ E ∗ we have:[ ∂, ω · f ]( η · e ) = ∂ (( − fη ω ∧ η · f ( e )) − ( − ω + f ( ω · f )( ∂ ( η ) · e )= ( − fη ∂ ( ω ) ∧ η · f ( e ) + ( − fη + ω ω ∧ ∂ ( η ) · f ( e ) − ( − ω + fη ω ∧ ∂ ( η ) · f ( e )= ( ∂ ( ω ) · f )( η · e ) . The composition product in A ∗ , ∗ X ( H om ∗O X ( E ∗ , E ∗ )) works in the following way: ω, η ∈ A ∗ , ∗ , f, g ∈ H om ∗O X ( E ∗ , E ∗ )( ω · f )( η · g ) = ( − fη ( ω ∧ η ) · f g , and the commutator is[ ω · f, η · g ] = ( ω · f )( η · g ) − ( − ( ω + f )( η + g ) ( η · g )( ω · f ) = ( − fη ( ω ∧ η ) · [ f, g ] . The above commutator and the differential [ δ + ∂, − ] = [ δ, − ] + ∂ give A ∗ , ∗ X ( H om ∗O X ( E ∗ , E ∗ )) astructure of DG-Lie algebra. Definition 2.1.
Let E ∗ be a finite complex of locally free sheaves on a complex manifold X . Aconnection of type (1 ,
0) on E ∗ is an element D of Hom C ( A ∗ , ∗ X ( E ∗ ) , A ∗ , ∗ X ( E ∗ )) such that:(1) D ( φ · s ) = dφ · s + ( − φ φ ∧ D ( s ), with s ∈ E r and φ ∈ A ∗ , ∗ X ;(2) if s ∈ E r is a holomorphic section then D ( s ) ∈ A , X ( E r ).The second condition is equivalent to D = D , + ∂ , with D , : A a,bX ( E r ) → A a +1 ,bX ( E r ) forevery a, b, r .Basically it is the same as giving a connection which is compatible with the holomorphicstructure on every vector bundle of the complex. In particular, connections of type (1 ,
0) alwaysexist.From now on, let D = D , + ∂ be a fixed connection of type (1 ,
0) on a fixed finite complex E ∗ of locally free sheaves. Denote by ∇ = [ D − ∂, − ] = [ D , , − ] : Hom ∗ C ( A ∗ , ∗ X ( E ∗ ) , A ∗ , ∗ X ( E ∗ )) → Hom ∗ C ( A ∗ , ∗ X ( E ∗ ) , A ∗ , ∗ X ( E ∗ )) ONNECTIONS AND L ∞ LIFTINGS OF SEMIREGULARITY MAPS 5 the adjoint operator of the (1 , D .As in the classical case, the adjoint operator [ D, − ] is a connection of type (1 ,
0) in the complexof locally free sheaves H om ∗O X ( E ∗ , E ∗ ). We need this fact only in the weaker form given by thefollowing lemma. Lemma 2.2.
In the above situation, for every h ∈ A ∗ , ∗ X ( H om ∗O X ( E ∗ , E ∗ )) we have [ D, h ] , ∇ ( h ) ∈ A ∗ , ∗ X ( H om ∗O X ( E ∗ , E ∗ )) . Proof.
Since every E i is locally free, we can describe A ∗ , ∗ X ( H om ∗O X ( E ∗ , E ∗ )) as the set of mor-phisms of graded sheaves h : A ∗ , ∗ X ( E ∗ ) → A ∗ , ∗ X ( E ∗ ) that are A ∗ , ∗ X linear, i.e., h ( f · s ) = ( − h f f · h ( s ), f ∈ A ∗ , ∗ X , s ∈ A ∗ , ∗ X ( E ∗ ).Thus, for every f ∈ A ∗ , ∗ X and s ∈ A ∗ , ∗ X ( E ∗ ) we have[ D, h ]( f s ) = ( − h f D ( f h ( s )) − ( − h h ( df · s + ( − f f · D ( s ))= ( − h f ( df · h ( s ) + ( − f f D ( hs )) − ( − h + h ( f +1) df · h ( s ) − ( − h + f + hf f · h ( D ( s ))= ( − ( h +1) f f · [ D, h ]( s ) . This proves that [
D, h ] ∈ A ∗ , ∗ X ( H om ∗O X ( E ∗ , E ∗ )). According to (2.1) we also have ∇ ( h ) = [ D, h ] − ∂ ( a ) ∈ A ∗ , ∗ X ( H om ∗O X ( E ∗ , E ∗ )) . (cid:3) Lemma 2.3.
In the above notation, define u = [ D, ∂ + δ ] = [ ∂ + δ, D ] = ∇ ( ∂ + δ ) , where the last equality follows by [ ∂, ∂ + δ ] = 0 . Then u ∈ A ∗ , ∗ X ( H om ∗O X ( E ∗ , E ∗ )) , and moreprecisely u ∈ A , X ( H om O X ( E ∗ , E ∗ )) ⊕ A , X ( H om O X ( E ∗ , E ∗ )) . Proof.
The element u : A ∗ , ∗ X ( E ∗ ) → A ∗ , ∗ X ( E ∗ ) is a morphism of graded sheaves of even degree andwe need to show that it is A ∗ , ∗ X -linear. By Lemma 2.2 we have [ D, δ ] ∈ A , X ( H om O X ( E ∗ , E ∗ ))and we only need to prove that [ D, ∂ ] is A ∗ , ∗ X -linear.For f ∈ A ∗ , ∗ X , s ∈ A ∗ , ∗ X ( E ∗ ) we have:[ D, ∂ ]( f s ) = D ( ∂f · s + ( − f f · ∂s ) + ∂ ( df · s + ( − f f D ( s ))= d ( ∂f ) · s + ( − f +1 ∂f · D ( s ) + ( − f df · ∂s + f D ( ∂ ( s ))+ ∂ ( df ) · s + ( − f +1 df · ∂s + ( − f ∂f · D ( s ) + f ∂ ( D ( s ))= f [ D, ∂ ]( s ) . Therefore [
D, ∂ ] ∈ A , X ( H om O X ( E ∗ , E ∗ )). (cid:3) Definition 2.4 (Atiyah class) . The Atiyah cocycle u ∈ A , ∗ X ( H om ∗O X ( E ∗ , E ∗ )) of the connection D of type (1 ,
0) as above is defined by the formula u = [ D, ∂ + δ ] = [ ∂ + δ, D ] = ∇ ( ∂ + δ ) . The fact that [ ∂ + δ, u ] = ∂ ( u ) + [ δ, u ] = 0, i.e., that u is a cocycle follows immediately byJacobi identity: [ ∂ + δ, u ] = [ ∂ + δ, [ ∂ + δ, D ]] = 12 [[ ∂ + δ, ∂ + δ ] , D ] = 0 . The Atiyah class of E ∗ is the cohomology class of the Atiyah cocycleAt( E ∗ ) = [ u ] ∈ H ( A , ∗ X ( H om ∗ ( E ∗ , E ∗ ))) = Ext X ( E ∗ , Ω ⊗ E ∗ ) . EMMA LEPRI AND MARCO MANETTI
The Atiyah class does not depend on the choice of the connection of type (1 , D, D ′ differ by a ∈ A , X ( H om O X ( E ∗ , E ∗ )), so that u ′ = [ D ′ , ∂ + δ ] = [ D + a, ∂ + δ ] = [ D, ∂ + δ ] + [ a, ∂ + δ ] = u + [ δ, a ] + ∂a, and u and u ′ represent the same cohomology class.It is straightforward to verify that the above definition of At( E ∗ ) is completely equivalent tothe one given in standard literature, especially [14, Section 10.1] and [7]: therefore the Atiyahclass is a homotopy invariant and depends only on the isomorphism class of of the complex E ∗ in the derived category of bounded complexes of locally free sheaves.In particular, if X is smooth projective and E ∗ → F is a finite locally free resolution of acoherent sheaf F , then Atiyah class of F is properly defined as At( F ) = At( E ∗ ) and dependsonly by the class of F in the bounded derived category of X . Lemma 2.5.
In the above setup, for every a ∈ A ∗ , ∗ X ( H om ∗O X ( E ∗ , E ∗ )) we have: [ δ + ∂, ∇ ]( a ) = [ u, a ] , [ ∇ ( ∂ ) , a ] = ∇ ( ∂a ) + ∂ ∇ ( a ) . In particular, if a is closed, then [ u, a ] is exact.Proof. We have[ u, a ] = [[ D , , δ + ∂ ] , a ] = [ D , , [ δ + ∂, a ]] + [ δ + ∂, [ D , , a ]] = [ δ + ∂, ∇ ]( a ) . while for the last equality, by (2.1),[ ∇ ( ∂ ) , a ] = [[ D , , ∂ ] , a ] = [ D , , [ ∂, a ]] + [ ∂, [ D , , a ]] = [ D − ∂, ∂a ] + ∂ [ D , , a ] = ∇ ( ∂a ) + ∂ ∇ ( a ) . (cid:3) The trace operator Tr : H om ∗O X ( E ∗ , E ∗ ) → O X can be extended toTr : A ∗ , ∗ X ( H om ∗O X ( E ∗ , E ∗ )) → A ∗ , ∗ X , Tr( ω · f ) = ω Tr( f ) . If we consider the DG-Lie algebra structure on A ∗ , ∗ X ( H om ∗ ( E ∗ , E ∗ )) given above, with differential[ δ, − ] + ∂ , and endow A ∗ , ∗ X with trivial bracket and differential ∂ , then the trace operator is amorphism of DG-Lie algebras, so thatTr([ δ, a ] + ∂a ) = ∂ Tr( a ) . Lemma 2.6.
In the above setup, for every h ∈ A ∗ , ∗ X ( H om ∗O X ( E ∗ , E ∗ )) we have Tr([
D, h ]) = d Tr( h ) .Proof. By linearity it is sufficient to consider the case h = η · g , with η ∈ A ∗ , ∗ X and g ∈H om n O X ( E ∗ , E ∗ ). It is clear that it is enough to consider g of degree 0, and by linearity wemay assume g concentrated in one degree, i.e., g = g l : E l → E l . Let e , . . . , e m be a local basisof holomorphic sections for E l , and let g ( e i ) = X j a ij e j , D ( e i ) = X j ω ij e j , Tr( g ) = ( − l X i a ii . Then d Tr( η · g ) = d ( η Tr( g )) = dη Tr( g ) + ( − η ηd Tr( g ) , ( ∇ + ∂ )( η · g )( e i ) = D ( X j ηa ij e j ) − ( − η ( η · g )( X j ω ij e j )= X j d ( η ) a ij e j + X j ( − η η ∧ d ( a ij ) e j + X j ( − η ηa ij ∧ D ( e j ) − ( − η X j η ∧ ω ij g ( e j )= X k d ( η ) a ik e k + X k ( − η η ∧ d ( a ik ) e k + X j,k ( − η ηa ij ∧ ω jk e k − ( − η X j,k η ∧ ω ij a jk e k . ONNECTIONS AND L ∞ LIFTINGS OF SEMIREGULARITY MAPS 7
ThereforeTr(( ∇ + ∂ )( η · g )) = ( − l X i dηa ii + ( − η ηd ( a ii ) + X j (cid:0) ( − η η ∧ ω ji a ij − ( − η η ∧ ω ij a ji (cid:1) = ( − l X i (cid:0) dηa ii + ( − η ηd ( a ii ) (cid:1) = dη Tr( g ) + ( − η ηd Tr( g ) . (cid:3) Remark . It is useful to note that since the trace is a morphism of differential graded Liealgebras, Tr([
D, h ]) = Tr(( ∇ + ∂ )( h )) = d Tr( h ) of Lemma 2.6 is equivalent toTr( ∇ ( h )) = ∂ Tr( h ) . Note also that if η ∈ A a,bX and g ∈ H om n O X ( E ∗ , E ∗ )), ∇ ( η · g ) ∈ A a +1 ,bX ( H om n O X ( E ∗ , E ∗ )).3. From connections and cyclic forms to L ∞ morphisms As in the previous section, let E ∗ : 0 → E p δ −→ E p +1 δ −→ · · · δ −→ E q → X . Definition 3.1.
By a cyclic (bilinear) form on the sheaf of DG-Lie algebras H om ∗O X ( E , E ) wemean a graded symmetric O X -bilinear product of degree 0 H om ∗O X ( E , E ) × H om ∗O X ( E , E ) h− , −i −−−−→ O X , such that h f, [ g, h ] i = h [ f, g ] , h i ∀ f, g, h . Equivalently, for every f, g, h we have h [ f, g ] , h i + ( − fg h g, [ f, h ] i = 0i.e., h− , −i is invariant under the adjoint action. In particular h [ δ, g ] , h i + ( − g h g, [ δ, h ] i = 0 , so the form is also [ δ, − ]-closed.Every cyclic form on H om ∗O X ( E , E ) has a natural extension A ∗ , ∗ X ( H om ∗O X ( E , E )) ⊙ h− , −i −−−−→ A ∗ , ∗ X , h φf, ψg i = ( − f ψ φ ∧ ψ h f, g i , and it is immediate to check that, for f, g ∈ A ∗ , ∗ X ( H om ∗O X ( E , E )):(3.1) ∂ h f, g i = h ∂f, g i + ( − f h f, ∂g i , and then h− , −i is ∂ + [ δ, − ] closed. Cyclic forms, have received a lot of attentions in severalrecent papers; for instance cyclic forms that are nondegerate in cohomology play a central rolein the proof of the formality conjecture for polystable sheaves on projective surfaces with torsioncanonical bundles, given in [5]. Definition 3.2.
We shall say that the a connection D of type (1 ,
0) on E ∗ is compatible withthe cyclic form h− , −i if h [ D, f ] , g i + ( − f h f, [ D, g ] i = d h f, g i , or equivalently if h∇ ( f ) , g i + ( − f h f, ∇ ( g ) i = ∂ h f, g i . for every f, g ∈ A ∗ , ∗ X ( H om ∗O X ( E , E )). EMMA LEPRI AND MARCO MANETTI
Example 3.3.
According to Lemma 2.6 and Remark 2.7, for every a, b ∈ C the form h f, g i = a Tr( f g ) + b Tr( f ) Tr( g )is a cyclic form of degree 0 compatible with every connection of type (1,0).We assume that the reader is familiar with the notion and basic properties DG-Lie algebrasand L ∞ morphisms between them, see e.g. [4, 5, 12, 13, 20, 26] and references therein. For thereader’s convenience and for fixing the sign convention, we only recall here the definition of L ∞ morphism of DG-Lie algebras in the version that we use for explicit computations.Let V be a graded vector space over a field of characteristic 0. Given v , . . . , v n homoge-neous vectors of V and a permutation σ of { , . . . , n } , we denote by χ ( σ ; v , . . . , v n ) = ± v σ (1) ∧ · · · ∧ v σ ( n ) = χ ( σ ; v , . . . , v n ) v ∧ · · · ∧ v n in the n th exterior power V ∧ n . We shall simply write χ ( σ ) instead of χ ( σ ; v , . . . , v n ) when thevectors v , . . . , v n are clear from the context. For instance if σ is the transposition exchanging 1and 2 we have χ ( σ ) = − ( − | v | | v | where | v | denotes the degree of the homogeneous vector v .Notice that if every v i has odd degree, then χ ( σ ) = 1 for every σ .Because of the universal property of wedge powers, we shall constantly interpret every linearmap V ∧ p → W as a graded skew-symmetric p -linear map V × · · · × V → W . Definition 3.4.
Let (
V, δ, [ − , − ]) and ( L, d, {− , −} ) be DG-Lie algebras over the same field. An L ∞ morphism g : V L is a sequence of linear maps g n : V ∧ n → L , n ≥
1, with g n of degree1 − n such that, g is a morphism of complexes, while for every n ≥ v , . . . , v n ∈ V homogeneous we have12 n − X p =1 X σ ∈ S ( p,n − p ) χ ( σ )( − (1 − n + p )( | v σ (1) | + ··· + | v σ ( p ) |− p ) n g p ( v σ (1) , . . . , v σ ( p ) ) , g n − p ( v σ ( p +1) , . . . , v σ ( n ) ) o + dg n ( v , . . . , v n ) = ( − n − X σ ∈ S (1 ,n − χ ( σ ) g n ( δ ( v σ (1) ) , v σ (2) , . . . , v σ ( n ) )+ ( − n − X σ ∈ S (2 ,n − χ ( σ ) g n − ([ v σ (1) , v σ (2) ] , v σ (3) , . . . , v σ ( n ) ) . Notice that the morphism of complexes g factors to a morphism g : H ∗ ( V ) → H ∗ ( L ), andthe above condition fon n = 2, which is equivalent to g ([ v , v ]) − { g ( v ) , g ( v ) } = dg ( v , v ) + g ( δv , v ) + ( − v g ( v , δv ) , tell us that g is a Lie morphism up to homotopy. In particular the map g : H ∗ ( V ) → H ∗ ( L ) isa morphism of graded Lie algebras.Conversely, given a morphism of graded Lie algebras τ : H ∗ ( V ) → H ∗ ( L ) we shall say thatan L ∞ morphism g : V L lifts τ if g induces τ in cohomology.In this paper we deal with L ∞ morphisms where the target L is an abelian DG-Lie algebra:this means that {− , −} = 0 and the above definition reduces to: Definition 3.5.
Let (
V, δ, [ − , − ]) be a DG-Lie algebra and ( L, d ) an abelian DG-Lie algebra.An L ∞ morphism g : V L is a sequence of maps g n : V ∧ n → L , n ≥
1, with g n of degree 1 − n such that the following conditions C n , n = 1 , , , . . . , are satisfied: C : g δ = dg ; C n , n ≥ : for every v , . . . , v n ∈ V homogeneous we have dg n ( v , . . . , v n ) = ( − − n X σ ∈ S (1 ,n − χ ( σ ) g n ( δv σ (1) , v σ (2) , . . . , v σ ( n ) )+ ( − − n X σ ∈ S (2 ,n − χ ( σ ) g n − ([ v σ (1) , v σ (2) ] , v σ (3) , . . . , v σ ( n ) ) . ONNECTIONS AND L ∞ LIFTINGS OF SEMIREGULARITY MAPS 9
Notice that if g n = 0 for every n ≥ N then C n is trivially satisfied for every n > N .We are now ready to prove the main result of this paper. In the following we consider theshifted quotient A ∗ , ∗ X A ≥ , ∗ X [2] of the de Rham complex by the 2nd subcomplex of the Hodge filtrationas a DG-Lie algebra with trivial bracket. Theorem 3.6.
Let E ∗ be a finite complex of locally free sheaves on a complex manifold X andlet h− , −i be a cyclic form of degree on H om ∗O X ( E ∗ , E ∗ ) which is compatible with a connection D of type (1 , . Then there is a L ∞ morphism between DG-Lie algebras over the field C g : A , ∗ X ( H om ∗O X ( E ∗ , E ∗ )) A ∗ , ∗ X A ≥ , ∗ X [2] with components g ( f ) = h u, f i = h f, u i ∈ A , ∗ X [2] ,g ( f, g ) = 12 (cid:16) h∇ ( f ) , g i − ( − fg h∇ ( g ) , f i (cid:17) ∈ A , ∗ X [2] ,g ( f, g, h ) = − h f, [ g, h ] i ∈ A , ∗ X [2] , and g n = 0 for every n > . As in the notation above, u = ∇ ( ∂ + δ ) is the Atiyah cocycle of theconnection D . Notice that the definition of g only involves the DG-Lie structure of A , ∗ X ( H om ∗O X ( E ∗ , E ∗ ))and not the associative composition product. Proof.
Since the theorem gives explicit formulas for the components g n , the proof reduces to astraightforward computation. Since g n = 0 for every n ≥ C n of Definition 3.5 for n = 1 , , ,
4. For C we have to prove that dg ( a ) = g ([ δ, a ] + ∂a ) . This follows from the fact that [ δ, u ] + ∂u = 0 and that on the subcomplex A , ∗ X ⊆ A ∗ , ∗ X A ≥ , ∗ X wehave d = ∂ : g ([ δ, a ] + ∂a ) = h u, [ δ, a ] + ∂a i = −h [ δ, u ] , a i + ∂ h u, a i − h ∂u, a i = −h [ δ, u ] + ∂u, a i + ∂ h u, a i = ∂ h u, a i = dg ( a ) . The condition C is g ([ δ, a ] + ∂a , a ) + ( − a g ( a , [ δ, a ] + ∂a ) = g ([ a , a ]) − dg ( a , a ) . On the left hand side we have g ([ δ, a ]+ ∂a , a ) + ( − a g ( a , [ δ, a ] + ∂a )= 12 (cid:16) h∇ ([ δ, a ]) , a i + h∇ ( ∂a ) , a i − ( − a a + a h∇ ( a ) , [ δ, a ] i− ( − a a + a h∇ ( a ) , ∂a i (cid:1) + 12 ( − a (cid:0) h∇ ( a ) , [ δ, a ] i + h∇ ( a ) , ∂a i− ( − a a + a h∇ ([ δ, a ]) , a i − ( − a a + a h∇ ( ∂a ) , a i (cid:17) = 12 (cid:16) h [ ∇ ( δ ) , a ]) , a i − h [ δ, ∇ ( a )]) , a i + h∇ ( ∂a ) , a i − ( − a a + a h∇ ( a ) , [ δ, a ] i− ( − a a + a h∇ ( a ) , ∂a i + ( − a h∇ ( a ) , [ δ, a ] i + ( − a h∇ ( a ) , ∂a i− ( − a a h [ ∇ ( δ ) , a ]) , a i + ( − a a h [ δ, ∇ ( a )]) , a i − ( − a a h∇ ( ∂a ) , a i (cid:17) = h∇ ( δ ) , [ a , a ] i + 12 (cid:16) h∇ ( ∂a ) , a i − ( − a a + a h∇ ( a ) , ∂a i + ( − a h∇ ( a ) , ∂a i− ( − a a h∇ ( ∂a ) , a i (cid:17) . On the right hand side, using Lemma 2.5, g ([ a , a ]) − dg ( a , a ) = h∇ ( δ ) + ∇ ( ∂ ) , [ a , a ] i − (cid:16) h ∂ ∇ ( a ) , a i− ( − a h∇ ( a ) , ∂a i − ( − a a h ∂ ∇ ( a ) , a i + ( − a a + a h∇ ( a ) , ∂a i (cid:17) = h∇ ( δ ) , [ a , a ] i − (cid:16) h ∂ ∇ ( a ) , a i − ( − a h∇ ( a ) , ∂a i − ( − a a h ∂ ∇ ( a ) , a i + ( − a a + a h∇ ( a ) , ∂a i (cid:17) + 12 (cid:16) h [ ∇ ( ∂ ) , a ] , a i − ( − a a h [ ∇ ( ∂ ) , a ] , a i (cid:17) = h∇ ( δ ) , [ a , a ] i + 12 (cid:16) − h ∂ ∇ ( a ) , a i + ( − a h∇ ( a ) , ∂a i + ( − a a h ∂ ∇ ( a ) , a i− ( − a a + a h∇ ( a ) , ∂a i + h∇ ( ∂a ) , a i − ( − a a h∇ ( ∂a ) , a i + h ∂ ∇ ( a ) , a i− ( − a a h ∂ ∇ ( a ) , a i (cid:17) = h∇ ( δ ) , [ a , a ] i + 12 (cid:16) ( − a h∇ ( a ) , ∂a i − ( − a a + a h∇ ( a ) , ∂a i + h∇ ( ∂a ) , a i − ( − a a h∇ ( ∂a ) , a i (cid:17) , so they are the same. For C we need to check that dg ( a , a , a ) = g ([ δ, a ] + ∂a , a , a ) − ( − a a g ([ δ, a ] + ∂a , a , a )+ ( − a ( a + a ) g ([ δ, a ] + ∂a , a , a ) − g ([ a , a ] , a ) + ( − a a g ([ a , a ] , a ) − ( − a ( a + a ) g ([ a , a ] , a ) . ONNECTIONS AND L ∞ LIFTINGS OF SEMIREGULARITY MAPS 11
Using the compatibility of the connection and the cyclic form, the terms involving g can beexpanded as: − g ([ a , a ] , a ) + ( − a a g ([ a , a ] , a ) − ( − a ( a + a ) g ([ a , a ] , a )= − (cid:0) h∇ ([ a , a ]) , a i − ( − a ( a + a ) h∇ ( a ) , [ a , a ] i (cid:1) + 12 ( − a a (cid:0) h∇ ([ a , a ]) , a i − ( − a ( a + a ) h∇ ( a ) , [ a , a ] i (cid:1) −
12 ( − a ( a + a ) (cid:0) h∇ ([ a , a ]) , a i − ( − a ( a + a ) h∇ ( a ) , [ a , a ] i (cid:1) = − (cid:16) h [ ∇ ( a ) , a ] , a i + ( − a h [ a , ∇ ( a )] , a i − ( − a ( a + a ) h∇ ( a ) , [ a , a ] i− ( − a a h [ ∇ ( a ) , a ] , a i − ( − a a + a h [ a , ∇ ( a )] , a i + ( − a a h∇ ( a ) , [ a , a ] i + ( − a ( a + a ) h [ ∇ ( a ) , a ] , a i + ( − a ( a + a )+ a h [ a , ∇ ( a )] , a i − h∇ ( a ) , [ a , a ] i (cid:17) = − ∂ h a , [ a , a ] i . On the other hand, g ([ δ, a ] + ∂a , a , a ) − ( − a a g ([ δ, a ] + ∂a , a , a ) + ( − a ( a + a ) g ([ δ, a ] + ∂a , a , a )= − (cid:0) h [ δ, a ] , [ a , a ] i + ( − a h a , [[ δ, a ] , a ] i + ( − a + a h a , [ a , [ δ, a ]] i + h ∂a , [ a , a ] i + ( − a h a , [ ∂a , a ] i + ( − a + a h a , [ a , ∂a ] i (cid:1) = − ∂ h a , [ a , a ] i so that we obtain dg ( a , [ a , a ]) = − d h a , [ a , a ] i = − ∂ h a , [ a , a ] i − ∂ h a , [ a , a ] i . Lastly, the condition C is g ([ a , a ] , a , a ) − ( − a a g ([ a , a ] , a , a ) + ( − a ( a + a ) g ([ a , a ] , a , a )+ ( − a ( a + a ) g ([ a , a ] , a , a ) − ( − a a + a a + a a g ([ a , a ] , a , a )+ ( − ( a + a )( a + a ) g ([ a , a ] , a , a ) = 0 . We have that12 h [ a , a ] , [ a , a ] i − ( − a a h [ a , a ] , [ a , a ] i + ( − a ( a + a ) h [ a , a ] , [ a , a ] i + ( − a ( a + a ) h [ a , a ] , [ a , a ] i − ( − a a + a a + a a h [ a , a ] , [ a , a ] i + ( − ( a + a )( a + a ) h [ a , a ] , [ a , a ] i = h [ a , a ] , [ a , a ] i − ( − a a h [ a , a ] , [ a , a ] i + ( − a ( a + a ) h [ a , a ] , [ a , a ] i = h a , [ a , [ a , a ]] i − ( − a a h a , [ a , [ a , a ]] i + ( − a ( a + a ) h a , [ a , [ a , a ]] i = h a , [ a , [ a , a ]] − ( − a a [ a , [ a , a ]] − [[ a , a ] , a ] i = 0 . (cid:3) Corollary 3.7.
Let E ∗ be a finite complex of locally free sheaves on a complex manifold X . Thenevery connection D of type (1 , on E ∗ gives an L ∞ morphism between DG-Lie algebras on the field C g : A , ∗ X ( H om ∗O X ( E ∗ , E ∗ )) A ∗ , ∗ X A ≥ , ∗ X [2] with components g ( f ) = − Tr( uf ) ∈ A , ∗ X [2] g ( f, g ) = −
12 Tr (cid:16) ∇ ( f ) g − ( − fg ∇ ( g ) f (cid:17) ∈ A , ∗ X [2] g ( f, g, h ) = 12 Tr( f [ g, h ]) ∈ A , ∗ X [2] , and g n = 0 for every n > .Proof. Use the cyclic form h f, g i = − Tr( f g ) in Theorem 3.6. (cid:3)
Remark . The homotopy class of the L ∞ morphism g of Corollary 3.7 depends on the choiceof the connection. This holds also for complex analytic connections, that is for connections wherethe Atiyah cocycle vanishes u = 0 and therefore g = 0. This implies that g factors to a bilineargraded skewsymmetric map in cohomology g : Ext iX ( E ∗ , E ∗ ) × Ext jX ( E ∗ , E ∗ ) → H i + j +1 A ∗ , ∗ X A ≥ , ∗ X ! that depends only on the homotopy class of g .In order to see that the above maps depends on the connection it is sufficient to consider theexample of a trivial bundle of rank 2 over an elliptic curve X . In this case, since Ω X is trivial,every complex analytic connection is of type D = d + θ , where θ is a 2 × H ( X, Ω X ) and then ∇ = ∂ + [ θ, − ]. Similarly Ext X ( E ∗ , E ∗ ) is identified with the Lie algebra M , ( C ) of 2 × g ( a, b ) = −
12 ([ θ, a ] b − [ θ, b ] a ) ∈ H (Ω X ) = H A ∗ , ∗ X A ≥ , ∗ X ! . If dz is a generator of H ( X, Ω X ) and θ = Cdz , with C ∈ M , ( C ), the conclusion follows byobserving that the rank of the bilinear map M , ( C ) × M , ( C ) → C , ( A, B )
12 Tr([
C, A ] B − [ C, B ] A ) = Tr( C [ A, B ]) , is equal to 0 when C is a multiple of the identity and is 2 otherwise.4. Semiregularity and deformations of coherent sheaves
Let F be a coherent sheaf on a complex manifold X admitting a locally free resolution(4.1) 0 → E − n δ −→ · · · δ −→ E → F → F is controlled by the DG-Lie algebra A , ∗ X ( H om ∗O X ( E ∗ , E ∗ ))defined in the previous sections for arbitrary finite complexes of locally free sheaves. This meansthat, over a local Artin C -algebra A , the deformations of F over A are determined by solutionsof the Maurer-Cartan equations ∂x + [ δ, x ] + 12 [ x, x ] = 0 , x ∈ M i ≥ A ,iX ( H om − i O X ( E ∗ , E ∗ ) ⊗ m A , with m A the maximal ideal of A . Moreover two solutions of the Maurer-Cartan equation giveisomorphic deformations if and only if they are gauge equivalent: when F is locally free this factis nowadays standard [12, 26] and extends quite easily to the general case, see e.g. [5, 11, 27]. ONNECTIONS AND L ∞ LIFTINGS OF SEMIREGULARITY MAPS 13
The resolution (4.1) can be also used for an explicit description of the (modified) semiregularitymap τ = X p ≥ τ p : Ext ∗ X ( F , F ) → M p ≥ H ∗ ( X, Ω ≤ pX [2 p ]) , τ ( a ) = Tr(exp( − At( F )) ◦ a ) . Since the quasi-isomorphic complexes E ∗ and F have the same Atiyah class we have At( F ) =[ u ], where u ∈ A , ∗ X ( H om ∗O X ( E ∗ , E ∗ )) is the Atiyah cocycle of a connection of type (1 ,
0) on E ∗ . Moreover the hypercohomology of Ω ≤ pX [2 p ] is computed by the truncated de Rham complex A ∗ , ∗ X /A >p, ∗ X [2 p ], and then every component τ p is induced in cohomology by the morphism ofcomplexes A , ∗ X ( H om ∗O X ( E ∗ , E ∗ )) → A ∗ , ∗ X A >p, ∗ X [2 p ] , f ( − p p ! Tr( u p f ) . Then, Corollary 3.7 immediately gives the following theorem.
Corollary 4.1.
Let F be a coherent sheaf on a complex manifold X admitting a locally freeresolution E ∗ . Then every connection of type (1 , on the resolution E ∗ gives a lifting of τ : Ext ∗ X ( F , F ) → H ∗ ( X, Ω ≤ X [2]) to an L ∞ morphism g : A , ∗ X ( H om ∗O X ( E ∗ , E ∗ )) A ∗ , ∗ X A > , ∗ X [2] . Now the application to deformation theory of Corollary 4.1 follows by a completely standardargument, the same used and explained, for instance, in [15, 20, 23, 24, 26]: every L ∞ morphism g : V L of DG-Lie algebras induces a morphism of deformation functors Def V → Def L (hereDef means the functor of Maurer-Cartan solution modulus gauge action) such that the inducedmap in cohomology commutes with obstruction maps. If L is abelian, then every obstruction inDef L is trivial, hence every obstruction in Def V belongs to the kernel of g : H ( V ) → H ( L ). Corollary 4.2.
Let F be a coherent sheaf on a complex manifold X admitting a locally freeresolution. Then every obstruction to the deformation of F belongs to the kernel of the map τ : Ext X ( F , F ) → H ( X, Ω ≤ X [2]) . If the Hodge to de Rham spectral sequence of X degenerates at E , then every obstruction to thedeformation of F belongs to the kernel of the map σ : Ext X ( F , F ) → H ( X, Ω X ) , σ ( a ) = − Tr(At( F ) ◦ a ) . Proof.
According to Corollary 4.1, the map τ lifts to an L ∞ morphism g : A , ∗ X ( H om ∗O X ( E ∗ , E ∗ )) A ∗ , ∗ X A ≥ , ∗ X [2]we have that the linear component g commutes with obstruction maps of associated deforma-tion functors. By construction the DG-Lie algebra A ∗ , ∗ X A ≥ , ∗ X [2] has trivial bracket and hence everyobstruction of the associated deformation functor is trivial.If the Hodge to de Rham spectral sequence of X degenerates at E then the inclusion ofcomplexes A , ∗ X [2] ⊂ A ∗ , ∗ X A ≥ , ∗ X [2] is injective in cohomology H ( X, Ω X ) ֒ → H ( X, Ω ≤ X [2])and the maps σ, τ have the same kernel. (cid:3) Outline of the analogous algebraic construction
Since the proof of the main theorem is mostly algebraic, it is not surprising the same can beslightly modified in order to have an algebraic analogue, valid on a smooth separated scheme X of finite type over a field K of characteristic 0, provided that an algebraic representative for R Hom( F , F ) is given. Here we give only a sketch of a possible algebraic proof, a more detaileddescription will be given in the forthcoming paper by the first author [22].Let E ∗ be a finite complex of locally free sheaves on X and let U = { U i } be an open affinecover of X . For simplicity of exposition we assume that E ∗ is a complex of free sheaves on every U i , although this additional assumption is unnecessary and it can be easily removed.Thus, according to [11], a possible DG-Lie algebra representing R Hom( F , F ) is the totaliza-tion Tot( U , H om ∗O X ( E ∗ , E ∗ )) of the cosimplicial DG-Lie algebra of ˇCech cochains in H om ∗O X ( E ∗ , E ∗ )with respect to the open cover U , see also [27]: here we follow the notation of [17, 26], where thereader can also find a complete and explicit definition of the totalization functorTot : { cosimplicial DG-vector spaces } → { DG-vector spaces } together with its main properties. Here we only recall that Tot preserves possible multiplicativestructures, hence transforms cosimplicial DG-Lie algebras (resp.: cosimplicial abelian DG-Liealgebras) into DG-Lie algebras (resp.: abelian DG-Lie algebras).Denote by Ω = Ω X/ K the sheaf of K¨ahler differentials. For every coherent sheaf M we denoteby D er K ( O X , M ) ≃ H om O X (Ω , M ) the sheaf of K -linear derivations O X → M , considered asa complex concentrated in degree 0. Following [17] we define the graded sheaf J ∗M = { ( f, α ) ∈ H om ∗ K ( E ∗ , M⊗E ∗ ) ×D er K ( O X , M ) | f ( ax ) = af ( x )+ α ( a ) ⊗ x, ∀ x ∈ E , a ∈ O X } . The same argument of [17] shows that J ∗M is a finite complex of coherent sheaves and thereexists a short exact sequence of complexes0 → H om ∗O X ( E ∗ , M ⊗ O X E ∗ ) f ( f, −−−−−→ J ∗M ( f,α ) α −−−−−→ D er K ( O X , M ) → . Then, a (germ of) algebraic connection on E ∗ may be conveniently defined as an element of J mapped onto the universal derivation d : O X → Ω. Clearly, since the map J → D er K ( O X , Ω)is generally not surjective on global sections, a global algebraic connection on E ∗ does notnecessarily exist.However, a global algebraic connection always exists in the totalization of J ∗ Ω with respect tothe affine open cover U . In fact, the exact sequence of coherent sheaves0 → H om ∗O X ( E ∗ , Ω ⊗ O X E ∗ ) f ( f, −−−−−→ J ∗ Ω ( f,α ) α −−−−−→ D er K ( O X , Ω) → U ; since Tot is an exact functor (see e.g. [8, 26]) we get an exact sequence0 → Tot( U , H om ∗O X ( E ∗ , Ω ⊗ O X E ∗ )) −−→ Tot( U , J ∗ Ω ) −−→ Tot( U , D er K ( O X , Ω)) → . In view of the natural inclusion of global sections into the totalization, the universal derivation d : O X → Ω belong to Tot( U , D er K ( O X , Ω)) and we may define a connection of type (1 , on E ∗ as an element D ∈ Tot( U , J ∗ Ω ) mapped onto d .Now everything works, mutatis mutandis, as in the previous sections: consider the complex O X [2] d −→ Ω[1] as a sheaf of abelian DG-Lie algebras and define the Atiyah cocycle u ∈ Tot( U , H om ∗O X ( E ∗ , Ω ⊗ O X E ∗ ))as the differential of D . Denote by ∇ = [ D, − ] : Tot( U , H om ∗O X ( E ∗ , E ∗ )) → Tot( U , H om ∗O X ( E ∗ , Ω ⊗ O X E ∗ ))the adjoint of D and use the same formulas of Theorem 3.6 in order to define an L ∞ morphism g : Tot( U , H om ∗O X ( E ∗ , E ∗ )) Tot( U , O X [2] d −→ Ω[1]) . ONNECTIONS AND L ∞ LIFTINGS OF SEMIREGULARITY MAPS 15
Finally, by Whitney’s integration theorem [13, 26] the cohomology of Tot( U , O X [2] d −→ Ω[1]) isthe same as the hypercohomology of O X [2] d −→ Ω[1] and the linear component g induces incohomology the first component τ of the (modified) semiregularity map. Acknowledgements.
We thank Francesco Meazzini and Ruggero Bandiera for useful discus-sions on the subject of this paper.
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